How can I make a texture flow along a bent mesh?
Edit: It looks like the problem has been fixed in the last version.

Old answer:

In 2.8 it looks like you need to mark seams and unwrap, then in the Shader Editor use the Texture Coordinate UV socket, play with the Location values of the Mapping node to make the texture move along the object. In the UV Editor you need to make sure that the UV is well placed if you want the texture to go in the right direction.

In 2.79 you can use the Texture Coordinate > Generated output socket instead of the Object output, then plug it into a Mapping node, play with the Mapping > Scale value to have the right proportion. Now with the Mapping > Location value you can make it move.